Efficient Exact Set-Similarity Joins

Efficient Exact Set-Similarity Efficient Exact Set-Similarity JoinsJoins

Arvind ArasuArvind ArasuVenkatesh GantiVenkatesh GantiRaghav KaushikRaghav Kaushik

DMX Group, Microsoft ResearchDMX Group, Microsoft Research

Sept. 15, 2006Sept. 15, 2006 Set-Similarity JoinsSet-Similarity Joins 22

Data CleaningData Cleaning

NameName StreetStreet CityCity StateState ZipZipINGRAM INGRAM MICROMICRO

1600 ST ANDREWS PL1600 ST ANDREWS PL SANTA ANASANTA ANA CACA 9279992799

GTE CORPGTE CORP 1 STAMFORD FORUM1 STAMFORD FORUM STAMFORDSTAMFORD CTCT

LOGISOFTLOGISOFT 274 GOODMAN ST N274 GOODMAN ST N ROCHESTERROCHESTER 1460714607

CIEDCCIEDC 1800 5TH ST1800 5TH ST LINCONLLINCONL ILIL 9279992799

INGRAM MCROINGRAM MCRO 1600 ST ANDREW’S 1600 ST ANDREW’S PLPL

SANTA ANASANTA ANA CACA 9279992799







CIEDCCIEDC 1800 5TH ST1800 5TH ST LINCOLINCONLNL ILIL 9279992799







GTE CORPGTE CORP 1 STAMFORD FORUM1 STAMFORD FORUM STAMFORDSTAMFORD CTCT 0690106901

LOGISOFTLOGISOFT 274 GOODMAN ST N274 GOODMAN ST N ROCHESTERROCHESTER NYNY 1460714607

CIEDCCIEDC 1800 5TH ST1800 5TH ST LINCOLINCOLNLN ILIL 9279992799


String Similarity JoinString Similarity Join

CITYCITY

ALABASTERALABASTER

ALBERTVILLEALBERTVILLE

……

……

……LINCOLNLINCOLN

……

……YUCAIPAYUCAIPA

Reference Table

…… …… CityCity …… ………… …… …… …… ……

…… …… LINCOLINCONLNL …… ……

…… …… …… …… ……

…… …… …… …… ……









String Similarity (Self) JoinString Similarity (Self) Join


Strings Strings Sets [CGK ’06] Sets [CGK ’06]

microsoft mcrosoft

{mc, cr, ro, os, so, of, ft}{mi, ic, cr, ro, os, so, of, ft}

(edit distance (edit distance ≤ 1) ----> (≤ 1) ----> (ΔΔ ≤ 4) ≤ 4)

2-grams2-grams

mcrosoft…

…

……

…

…

…

microsoft…

…

……

…

…

… SR

String Sim Join edit distance edit distance ≤ 1≤ 1

Strings Sets

mcrosoft…

…

……

…

…

…

microsoft…

…

……

…

…

…

Set Sim Join ΔΔ ≤ 4≤ 4

R S

TokenizeTokenize

Post-Process

Strings Sets


String String Set: Advantages Set: Advantages

Generalizes to many string similarity Generalizes to many string similarity funcsfuncs Powerful primitivePowerful primitive

Sets Sets ≈ Relations≈ Relations Leverage relational data processingLeverage relational data processing

[CGK ‘06][CGK ‘06]


ContributionsContributions

New algorithms for set-similarity New algorithms for set-similarity joinsjoins Exact answersExact answers Performance guaranteesPerformance guarantees Outperform previous exact algorithmsOutperform previous exact algorithms

Orders of magnitudeOrders of magnitude

Exact answers are important for operatorsExact answers are important for operators


OutlineOutline

IntroductionIntroduction AlgorithmsAlgorithms ExperimentsExperiments ConclusionConclusion

{ mi, ic, cr, ro, os, so, of, ft }

{ lo, og, gi, is, so, of, ft }

{ … }

{ … }

{ … }

{ … }

{ bo, oe, ei, in, ng }{ mc, cr, ro, os, so, of, ft }

{ lg, gi, is, so, of, ft }

{ … }

{ … }

{ … }

{ … }

{ … }

SR



{ … }

{ … }

{ … }

{ … }

{ bo, oe, ei, in, ng }{ mc, cr, ro, os, so, of, ft }


{ … }

{ … }

{ … }

{ … }

{ … }

SR

Intersection size Intersection size ≥ 5 ≥ 5



{ … }

{ … }

{ … }

{ … }

{ bo, oe, ei, in, ng }

{ mc, cr, ro, os, so, of, ft }


{ … }

{ … }

{ … }

{ … }

{ … }

SR






{ … }

{ … }

{ … }

{ … }




{ … }

{ … }

{ … }

{ … }

{ … }

SR






{ … }

{ … }

{ … }

{ … }


{ … }

{ … }

{ … }

{ … }

{ … }

SR



Sim Sim ( ( rrii , s , sjj ) ) ≥ ≥ θθ



ss22

ss33

ssmm

ss11

rr22

rr33

rrnn

rr11

{ … }

{ … }

{ … }

{ … }


{ … }

{ … }

{ … }

{ … }

{ … }

SR



Sim Sim ( ( rrii , s , sjj ) ) ≥ ≥ θθ



ss22

ss33

ssmm

ss11

rr22

rr33

rrnn

rr11

Larg

e

Input:Input: R: R: rr11 , , rr22 , … , , … , rrnn (n sets) (n sets) S: S: ss1 1 , , ss2 2 , … , , … , ssmm (m sets) (m sets)

Output: All pairs (Output: All pairs (rrii , s , sj j ) such that:) such that: ||rrii ΔΔ s sjj | | ≤ ≤ kk

Set-Similarity Join: Symmetric Set-Similarity Join: Symmetric DifferenceDifference

≤ kRunning example: Running example: k k = 4= 4


Alternate Set Alternate Set RepresentationRepresentation

s = { 4, 10, 13, 24, 29, 35, 41, 46, 48 }



s = { 4, 10, 13, 24, 29, 35, 41, 46, 48 }

1 25 50


EnumerationEnumeration

s

r

|r Δ s | ≤ 4



s

r

|r Δ s | ≤ 4



s

r

|r Δ s | ≤ 4

ErrorsErrors



2 3 4 51

s

r

|r Δ s | ≤ 4


Enumeration: Signature Enumeration: Signature GenerationGeneration

s

, , ,,{ }

Sig (s )



s

, , ,,{ }

Sig (s )

{ 0x4f72ba91, 0x29c8af10, 0x594b2c17, 0xa3b0e20f, 0xdd21f32a}

Hash32()


Property of SignaturesProperty of Signatures

||r r ΔΔ ss | | ≤ 4≤ 4 Sig (Sig (rr ) Sig ( ) Sig (s s ) ) ≠ ≠ ΦΦ

UU

2 3 4 51

s

r


Enumeration: AlgorithmEnumeration: Algorithm

Generate signatures for each Generate signatures for each rrii , , ssjj

Enumerate (Enumerate (rrii , s , sjj ) s.t ) s.t Sig ( Sig (rrii ) Sig () Sig (ssjj ) ) ≠ ≠ ΦΦ

Output those satisfying |Output those satisfying |rrii ΔΔ ssjj | ≤ 4| ≤ 4

U



s1

s5

s2

s3

s4

Sig (s2)

Sig (s5)

Sig (s3)

Sig (s4)UU

r1

r5

r2

r3

r4

Sig (s1)

Sig (r2)

Sig (r5)

Sig (r3)

Sig (r4)

Sig (r1)

Sig (Sig (rr22)) Sig (Sig (ss11)) ≠≠ ΦΦ



s1

s5

s2

s3

s4

Sig (s2)

Sig (s5)

Sig (s3)

Sig (s4)UU

r1

r5

r2

r3

r4

Sig (s1)

Sig (r2)

Sig (r5)

Sig (r3)

Sig (r4)

Sig (r1)




s1

s5

s2

s3

s4

Sig (s2)

Sig (s5)

Sig (s3)

Sig (s4)UU

r1

r5

r2

r3

r4

Sig (s1)

Sig (r2)

Sig (r5)

Sig (r3)

Sig (r4)

Sig (r1)


OutputOutput False positive candidate pairsFalse positive candidate pairs

S (Id, Elem)

R.Sig = S.Sig

δ R.Id, S.Id

R (Id, Elem)

Post-Process each R.Id, S.Id

Gen SignaturesGen Signatures

S’ (Id, Sig)R’ (Id, Sig)


No False Positive Candidate No False Positive Candidate PairPair

2 3 4 51

s

r

|r Δ s | = 5


False Positive Candidate False Positive Candidate PairPair

s2

s1

2 3 4 51

|r Δ s | = 5


Enumeration: PerformanceEnumeration: Performance

0

0.25

0.5

0.75

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Symmetric Difference

Pro

bab

ility

of

Co

mm

on

Sig

nat

ure

k = 4



0

0.25

0.5

0.75

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Pro

bab

ility

of

Co

mm

on

Sig

nat

ure

Ideal PerformanceIdeal Performance

k = 4



|r Δ s | ≤ 4

s

r



2 3 4 61 5

s

r

|r Δ s | ≤ 4



s1

2 3 4 61 5



s1

2 3 4 61 5

( )( )6622

= 15= 15


AlgorithmAlgorithm

Generate signatures for each Generate signatures for each rrii , , ssjj

Enumerate (Enumerate (rrii , s , sjj ) s.t ) s.t Sig ( Sig (rrii ) Sig () Sig (ssjj ) ) ≠ ≠ ΦΦ

Output those satisfying |Output those satisfying |rrii ΔΔ ssjj | ≤ 4| ≤ 4

U

Only the signature function changesOnly the signature function changes



0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Pro

b. o

f Com

mon

Sig

natu

re

n1 = 5 n1 = 6

k = 4


False Positive Candidate False Positive Candidate PairPair

2 3 4 61 5

s

r

|r Δ s | = 5



0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Prob

. of C

omm

on S

igna

ture

n1 = 5 n1 = 6 n1 = 7 n1 = 20

k = 4



0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Prob

. of C

omm

on S

igna

ture

n1 = 5 n1 = 6 n1 = 7 n1 = 20

55

15153535

48454845

k = 4


PartEnum: Divide and PartEnum: Divide and ConquerConquer

s1

21

k = 4

k2 = 1k1 = 2

Generate signatures using EnumerationGenerate signatures using Enumeration


PartEnum: Asymptotic PartEnum: Asymptotic PerformancePerformance

Theorem: There is an instance of Theorem: There is an instance of PartEnum such that: PartEnum such that: If If ||r r ΔΔ s s || > 7.5 > 7.5 kk, , then then r r and and s s do not do not

share a signature with probability 1 – share a signature with probability 1 – o(1)o(1)

The number of signatures per set: The number of signatures per set: O (O (kk22 ) )


PartEnum: SummaryPartEnum: Summary

Set-Similarity Joins with predicate Set-Similarity Joins with predicate ||rr ΔΔ ss | ≤ | ≤ kk

Theoretical guaranteesTheoretical guarantees First exact algorithmFirst exact algorithm


Other resultsOther results

PartEnum extensions:PartEnum extensions: Larger class of set-similarity join predicatesLarger class of set-similarity join predicates

JaccardJaccard Basic idea: reduce to symmetric set Basic idea: reduce to symmetric set

differencedifference WtEnumWtEnum class of signature functions: class of signature functions:

Use frequency of elementsUse frequency of elements Weighted set-similarity joinsWeighted set-similarity joins


OutlineOutline

IntroductionIntroduction AlgorithmsAlgorithms ExperimentsExperiments ConclusionConclusion

S (Id, Elem)

R.Sig = S.Sig

δ R.Id, S.Id

R (Id, Elem)



Implementation

DBMSDBMS

Client + DBMSClient + DBMS

DBMSDBMS

ClientClient


Previous WorkPrevious Work

Prefix Filtering [CGK ’06]Prefix Filtering [CGK ’06] ExactExact

Locality Sensitive Hashing [IM ’98]Locality Sensitive Hashing [IM ’98] ApproximateApproximate False negative rate: 5%False negative rate: 5%


Data SetsData Sets

Organization addresses [MS Sales]Organization addresses [MS Sales] Concatenation: Org name, street, city, Concatenation: Org name, street, city,

zipzip Input size: 1 millionInput size: 1 million Avg. length: 11 words, 58 charsAvg. length: 11 words, 58 chars Tokenization: Words, n-gramsTokenization: Words, n-grams


Jaccard, 1M, MS SalesJaccard, 1M, MS Sales

0

1000

2000

3000

4000

PEN LSH PF PEN LSH PF PEN LSH PF

Sec

on

ds

SigGen CandPair PostFilter

0.80.9 0.85

S (Id, Elem)

R.Sig = S.Sig

δ R.Id, S.Id

R (Id, Elem)



Evaluation

DBMSDBMS

DBMSDBMS

IntermediateIntermediateResult sizeResult size

Client + DBMSClient + DBMS

ClientClient

Jaccard, 1M, MS SalesJaccard, 1M, MS Sales

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

PEN LSH PF PEN LSH PF PEN LSH PF

Inte

rmed

iate

Res

ult

Siz

e

0.80.9 0.85


Jaccard, SyntheticJaccard, Synthetic

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1.0E+10

1.0E+11

1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09

Input Size

Inte

rmed

iate

Res

ult S

ize

LSH(0.95) PEN PF


Similar Results for …Similar Results for …

Other data setsOther data sets DBLP, Synthetic data setsDBLP, Synthetic data sets

Other similarity functionsOther similarity functions Weighted jaccardWeighted jaccard Edit distanceEdit distance


ConclusionConclusion

New algorithms for set-similarity New algorithms for set-similarity joinsjoins ExactExact Performance guaranteesPerformance guarantees Outperform previous exact algorithmsOutperform previous exact algorithms

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Efficient Exact Set-Similarity Joins

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